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1917]
AT THE SUEFACE OF DEEP WATER
483
first.    If the series are to be depended upon, we must clearly take a smaller value.    I have chosen a = -fa, and this makes by (15), (18), (20)
£ = - -000,052,42,    7 = '000,000,976,   g = "989,736,92 ....... (21)*
The next step is the calculation of approximate values of y from (11), which now takes the form
y = ~ '0051 + 101,165,0 cos os
- '005,183,3 cos 2a? + '000,399,6 cos 3a?
- -000,033,3 cos 4« + '000,003,3 cos ox ................ (22)
For example, when an = 0, y = '091,251,3. The values of y calculated from (22) at steps of 22|° (as in Burnside's work) are shown in column 2 of Table I.
We have next to examine how nearly the value of y afforded by (22) really makes i/r vanish, and if necessary to calculate corrections. To this 8 and e in (1) do not contribute sensibly and we find ty = + '000,015,4 for a? = 0. In order to reduce i|r to zero, we must correct the value of y. With sufficient approximation we have in general
ty  % (1 + TO- e~y cos *')f; or in the present case
so that the corrected value of y for x = 0 is '091,237,2.    If we repeat the calculation, using the new value of y, we find -^ = 0.
TABLE I.J
.T	y from (22)	y corrected	U*-2gy-l	Corrected by dp
0	+ '091,251,3   '	+ 091,237,2	010,104,9	45
22*	+ -084,839,7	+ -084,841,9	. 4,7	44
45	+ '066,182,8	+ 066,181,8	. 4,3	43
67*	+ 036,913,1	+ 036,915,1	. 4,1	44
90	+ -000,050,0	+ -000,052,4	. 4,2	46
112*	- -039,782,7	- -039,780,2	. 4,4	47
135	-076,316,2	- -076,317,5	. 4,3	43
157*	- -102,381,1	- -102,395,1	. 4,7	44
180	-111,884,7	-111,907,9	010,105,1	47
[* With the corrections specified in the footnote on p. 482 \ve have 7 = -000,000,905, g = -989,737,42. W. P. S.]
t The double use of S will hardly cause confusion.
[J With the corrections specified in the footnotes on pp. 481, 482, and calculating direct from (2), with the inclusion of the term 65e~4w cos 4#, I find that the first 5 figures in the value of I72 - 2<jy -1 are as in the table, whilst the last 2 figures, proceeding in order from £p=0 to a; = 180, become 45, 45, 44, 43, 42, 42, 45, 51, 53; after making 6 modifications in "y corrected" (third column), the first 6 figures of which remain as printed, whilst the last becomes, taken in the same order, 1, 9, 9, 1, 4, 3, 6, 3, 8, these modified values of y in every case reducing \j/ to zero to 7 places of decimals. W. F. S.]
312s X  CL cos'x; ........................... (8)